TeXipedia

ncong

Indicates non-congruence between mathematical expressions or geometric figures, serving as the negation of the congruence relation.

Overview

Essential in mathematical proofs and geometric reasoning where demonstrating non-congruence is required.

  • Common in abstract algebra for showing relationships between algebraic structures
  • Used in geometry to express that shapes or figures do not have the same size and shape
  • Appears frequently in number theory when discussing modular arithmetic
  • Helpful in educational contexts when teaching equivalence relationships and their negations

Examples

Showing that two triangles are not congruent based on their side lengths.

ABCDEF\triangle ABC \ncong \triangle DEF
\triangle ABC \ncong \triangle DEF

Demonstrating non-congruence of remainders in modular arithmetic.

52(mod3)5 \ncong 2 \pmod{3}
5 \ncong 2 \pmod{3}

Expressing that two algebraic expressions are not congruent under a specific modulus.

x2+1x+1(mod2)x^2 + 1 \ncong x + 1 \pmod{2}
x^2 + 1 \ncong x + 1 \pmod{2}